Linear Algebra

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Theorems

solution Find the eigenvalues of the matrix {\begin{bmatrix}5&2\\3&6\\\end{bmatrix}}
solution Define the adjoint of a matrix.
solution Define a self-adjoint matrix.
solution Define a unitary matrix.
solution Show that (A^{*})^{*}=A\,
solution Show that (AB^{*})^{*}=BA^{*}\,
solution Show that AA^{*}\, is self-adjoint.
solution Show that the identity matrix I\, is self-adjoint.
solution Show that the zero matrix {\mathcal  {O}}\, is self-adjoint.
solution Show that (\alpha A+\beta B)^{*}=(\overline {\alpha }A^{*}+\overline {\beta }B^{*})\,
solution Let A\, be an n\times n matrix such that A^{2}=A\,, and let I\, be the n\times n identity matrix. Prove that {{\rm {rank}}}(A)+{{\rm {rank}}}(A-I)=n\,.
solution Let A\, be an m\times n matrix. Prove that {{\rm {Null}}}(A)^{\perp }={{\rm {Col}}}(A^{T}).

Matrices

Basic Problems

solution If A={\begin{bmatrix}0&1&2\\2&3&4\end{bmatrix}}\, and B={\begin{bmatrix}1&0&0\\2&-3&1\end{bmatrix}}\, evaluate 2A+3B\,

solution Find x\, such that {\begin{bmatrix}1&x&1\end{bmatrix}}{\begin{bmatrix}1&3&2\\2&5&1\\15&3&2\end{bmatrix}}{\begin{bmatrix}1\\2\\x\end{bmatrix}}=0\,.

solution If A={\begin{bmatrix}2&3\\-1&2\end{bmatrix}}\, Show that A^{2}-4A+7I=O\,

Solution If w is cube root of unity,show that {{\begin{bmatrix}1&w&w^{2}\\w&w^{2}&1\\w^{2}&1&w\end{bmatrix}}+{\begin{bmatrix}w&w^{2}&1\\w^{2}&1&w\\w&w^{2}&1\end{bmatrix}}}{\begin{bmatrix}1\\w\\w^{2}\end{bmatrix}}={\begin{bmatrix}0\\0\\0\end{bmatrix}}\,

solution If A={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}\, for all integral values of n,show that A^{n}={\begin{bmatrix}\cos n\theta &\sin n\theta \\-\sin n\theta &\cos n\theta \end{bmatrix}}\,

solution Find the value of determinant of the matrix {\begin{bmatrix}29&26&22\\25&31&27\\63&54&46\end{bmatrix}}\,

solution Show that {\begin{vmatrix}bc&b+c&1\\ca&c+a&1\\ab&a+b&1\end{vmatrix}}=(a-b)(b-c)(c-a)\,

solution Prove that the determinant of the matrix {\begin{bmatrix}y+z&z&y\\z&z+x&x\\y&x&x+y\end{bmatrix}}=4xyz\,

solution Show that {\begin{vmatrix}a+b&b+c&c+a\\b+c&c+a&a+b\\c+a&a+b&b+c\end{vmatrix}}=2{\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}}\,

solution Without expanding the determinant of the matrix A={\begin{bmatrix}0&p-q&p-r\\q-p&0&q-r\\r-p&r-q&0\end{bmatrix}}\, prove that |A|=0\,

solution Prove that {\begin{vmatrix}-2a&a+b&c+a\\b+a&-2b&b+c\\c+a&c+b&-2c\end{vmatrix}}=4(a+b)(b+c)(c+a)\,

solution If a,b,c are distinct, abc\not \equiv 0\, and {\begin{vmatrix}a&a^{3}&a^{4}-1\\b&b^{3}&b^{4}-1\\c&c^{3}&c^{4}-1\end{vmatrix}}=0\, then prove that abc(ab+bc+ca)=a+b+c\,

solution Show that {\begin{vmatrix}a&a+b&a+b+c\\2a3a+2b&4a+3b+2c\\3a&6a+3b&10a+6b+3c\end{vmatrix}}=a^{3}\,

solution Prove that {\begin{vmatrix}a^{2}&bc&ac+c^{2}\\a^{2}+ab&b^{2}&ac\\ab&b^{2}+bc&c^{2}\end{vmatrix}}=4a^{2}b^{2}c^{2}\,

solution Without expanding the determinant prove that {\begin{vmatrix}1&bc&b+c\\a&ca&c+a\\1&ab&a+b\end{vmatrix}}={\begin{vmatrix}1&a&a^{2}\\1&b&b^{2}\\1&c&c^{2}\end{vmatrix}}\,

solution Solve for x given {\begin{vmatrix}x-2&2x-3&3x-4\\x-4&2x-9&3x-16\\x-8&2x-27&3x-64\end{vmatrix}}=0\,

solution Show that the determinant of the matrix {\begin{bmatrix}\cos(\theta +\alpha )&\sin(\theta +\alpha )&1\\\cos(\theta +\beta )&\sin(\theta +\beta )&1\\\cos(\theta +\alpha )&\sin(\theta +\alpha )&1\end{bmatrix}}\, is independent of theta.

solution Show that {\begin{vmatrix}b+c&a-b&a\\c+a&b-c&b\\a+b&c-a&c\end{vmatrix}}=3abc-a^{3}-b^{3}-c^{3}\,


Inverse & Rank of a Matrix

solution If A and B are two non-singular matrices of the same type,then adjoint(AB)=(adjoint B)(adjoint A)

solution If A,B are invertible matrices of the same order,then (AB)^{{-1}}=B^{{-1}}A^{{-1}}\,

Solution Compute the adjoint of the matrix A={\begin{bmatrix}1&2&2\\2&3&0\\0&1&2\end{bmatrix}}\,

solution Find the adjoint and inverse of A={\begin{bmatrix}2&3&4\\4&3&1\\1&2&4\end{bmatrix}}\,

solution Determine the rank of A={\begin{bmatrix}4&2&3\\8&4&6\\-2&-1&-1.5\end{bmatrix}}\,

solutionFind the rank of A,rank of B A={\begin{bmatrix}1&5&4\\0&3&2\\2&3&10\end{bmatrix}}\,, B={\begin{bmatrix}1&1&1\\2&2&2\\3&3&3\end{bmatrix}}\,

solution Determine the values of b such that the rank of the matrix A is 3. A={\begin{bmatrix}1&1&-1&0\\4&4&-3&1\\b&2&2&2\\9&9&b&3\end{bmatrix}}\,

solution Find the non-singular matrices P and Q such that the normal form of A is PAQ where A={\begin{bmatrix}1&3&6&-1\\1&4&5&1\\1&5&4&3\end{bmatrix}}\,. Hence find its rank.

solution Find P and Q such that the normal form of A={\begin{bmatrix}1&-1&-1\\1&1&1\\3&1&1\end{bmatrix}}\, is PAQ. Hence the find the rank.

solution A={\begin{bmatrix}1&5&4\\0&3&2\\2&3&10\end{bmatrix}}\,, B={\begin{bmatrix}1&1&1\\2&2&2\\3&3&3\end{bmatrix}}\,. Find the rank of A+B\, and AB\,.

solution Solve by Cramer's rule x+y+z=11,2x-6y-z=0,3x+4y+2z=0\,

Inner Products

solution Define an inner product.
solution Show that \langle x,\alpha y\rangle =\overline {\alpha }\langle x,y\rangle
solution Show that \langle x,y+z\rangle =\langle x,y\rangle +\langle x,z\rangle
solution Show that \langle x,y\rangle +\langle y,x\rangle =2{\mbox{ Re }}\langle x,y\rangle
solution Show that \langle x,y\rangle -\langle y,x\rangle =2i{\mbox{ Im }}\langle x,y\rangle
solution Show that \langle \alpha x,\beta y\rangle =\alpha \overline {\beta }\langle x,y\rangle
solution Show that \langle \alpha x,\alpha y\rangle =|\alpha |^{2}\langle x,y\rangle
solution Show that \langle -x,-y\rangle =\langle x,y\rangle
solution Show that \langle x,{\vec  {0}}\rangle =0
solution Show that \langle {\vec  {0}},y\rangle =0
solution Show that \langle x,x\rangle is always real.

Vector Algebra

Vector Addition

solution If AB,BE,CF\, are the medians of a triangle,then prove that \quad {\bar  {AB}}+{\bar  {BE}}+{\bar  {CF}}={\bar  {O}}\,

solution If G is the centroid of the triangle ABC,prove that {\bar  {GA}}+{\bar  {GB}}+{\bar  {GC}}={\bar  {O}}\, where A,B,C\, are the vertices of the triangle ABC and {\bar  {O}}\, is the point vector

solution The position vectors of A and B are 2{\bar  {i}}+{\bar  {j}}-{\bar  {k}},{\bar  {i}}+2{\bar  {j}}+3{\bar  {k}}\, respectively.Find the position vector of the point which divides the line segment AB in the ration 2:3.

solution If {\bar  {a}}=2{\bar  {i}}+{\bar  {k}},{\bar  {b}}=3{\bar  {i}}+4{\bar  {k}},{\bar  {c}}=8{\bar  {i}}+9{\bar  {k}}\, then express {\bar  {c}}\, as a linear combination of {\bar  {a}}\, and {\bar  {b}}\,.

solution If {\bar  {OA}}={\bar  {i}}+{\bar  {j}}+{\bar  {k}},{\bar  {AB}}=3{\bar  {i}}+2{\bar  {j}}+{\bar  {k}},{\bar  {BC}}={\bar  {i}}+2{\bar  {j}}-2{\bar  {k}},{\bar  {CD}}=2{\bar  {i}}+{\bar  {j}}+3{\bar  {k}}\,,then find the position vector of D.

solution If {\bar  {a}}\, is the position vector whose point is (3,-2)\,.Find the coordinates of a point B such that {\bar  {AB}}={\bar  {a}}\,,the coordinates of A are (-1,5)\,

solution Find a vector of magnitude 6units which is parallel to the vector {\bar  {i}}+{\sqrt  {3}}{\bar  {j}}\,

solution Find the magnitude of the vector 7{\bar  {i}}-3{\bar  {j}}+5{\bar  {k}}\,

solution If the position vectors of A and B are 2{\bar  {i}}-9{\bar  {j}}-4{\bar  {k}},6{\bar  {i}}-3{\bar  {j}}+8{\bar  {k}}\, respectively,find the unit vector in the direction of AB.

solution If the position vectors of A and B are {\bar  {i}}+3{\bar  {j}}-7{\bar  {k}},5{\bar  {i}}-2{\bar  {j}}+4{\bar  {k}}\, respectively,determine the direction cosines of {\bar  {AB}}\,

solution In a triangle ABC if A=2{\bar  {i}}+4{\bar  {j}}-{\bar  {k}},B=4{\bar  {i}}+5{\bar  {j}}+{\bar  {k}},C=3{\bar  {i}}+6{\bar  {j}}-3{\bar  {k}}\, and D is the mid point of the side BC, then find the length of AD.

solution Show that the points represented by {\bar  {i}}+2{\bar  {j}}+3{\bar  {k}},3{\bar  {i}}+4{\bar  {j}}+7{\bar  {k}},-3{\bar  {i}}-2{\bar  {j}}-5{\bar  {k}}\, are collinear.

solution Show that the points A,B,C,D with position vectors 6{\bar  {i}}-7{\bar  {j}},16{\bar  {i}}-19{\bar  {j}}-4{\bar  {k}},3{\bar  {j}}-6{\bar  {k}},2{\bar  {i}}+5{\bar  {j}}+10{\bar  {k}}\, are not coplanar.

solution Prove that three points whose vectors are {\bar  {i}}+2{\bar  {j}}+3{\bar  {k}},-{\bar  {i}}-{\bar  {j}}+8{\bar  {k}},4{\bar  {i}}+4{\bar  {j}}+6{\bar  {k}}\, form an equilateral triangle.

solution Show that the triangle ABC whose vertices are 7{\bar  {i}}+10{\bar  {k}},-{\bar  {i}}+6{\bar  {j}}+6{\bar  {k}},-4{\bar  {i}}+9{\bar  {j}}+6{\bar  {k}}\, is isoscles and right angled.

solution Obtain the point of intersection of the line joining the points {\bar  {i}}-2{\bar  {j}}-{\bar  {k}},2{\bar  {i}}+3{\bar  {j}}+{\bar  {k}}\, with the plane through the points 2{\bar  {i}}+{\bar  {j}}-3{\bar  {k}},4{\bar  {i}}-{\bar  {j}}+2{\bar  {k}}\, and 3{\bar  {i}}+{\bar  {k}}\,

Vector Product

solution Show that the points whose position vectors are 2{\bar  {i}}-{\bar  {j}}+{\bar  {k}},{\bar  {i}}-3{\bar  {j}}-5{\bar  {k}},3{\bar  {i}}-4{\bar  {j}}-4{\bar  {k}}\, are the vertices of a right angled triangle.

solution Find a vector {\bar  {d}}\, which is perpendicular to both {\bar  {a}}=4{\bar  {i}}+5{\bar  {j}}-{\bar  {k}},{\bar  {b}}={\bar  {i}}-4{\bar  {j}}+5{\bar  {k}}\, and {\bar  {d}}\cdot {\bar  {c}}=21\, where{\bar  {c}}=3{\bar  {i}}+{\bar  {j}}-{\bar  {k}}\,

solution If {\bar  {a}},{\bar  {b}},{\bar  {c}}\, are mutually perpendicular vectors of equal magnitude,show that {\bar  {a}}+{\bar  {b}}+{\bar  {c}}\, is equally inclined to {\bar  {a}},{\bar  {b}},{\bar  {c}}\,

solution Dot products of the vectors {\bar  {i}}+{\bar  {j}}-3{\bar  {k}},{\bar  {i}}+3{\bar  {j}}-2{\bar  {k}},2{\bar  {i}}+{\bar  {j}}+4{\bar  {k}}\, are 0,5,8\, respectively.Find the vector.

solution Find the vector equation of a plane which is at a distance of 5units from the origin and which has 2{\bar  {i}}+3{\bar  {j}}+6{\bar  {k}}\, as a normal vector.

solution Find the vector equation of the plane through the point A(3,-2,1)\, and perpendicular to the vector 4{\bar  {i}}+7{\bar  {j}}-4{\bar  {k}}\,.

solution Find the equation of the plane passing through the point (3,4,5)\, and parallel to the plane {\bar  {r}}\cdot (2{\bar  {i}}+3{\bar  {j}}-{\bar  {k}})=6\,

solution If {\bar  {a}}=2{\bar  {i}}-3{\bar  {j}}+5{\bar  {k}},{\bar  {b}}=-{\bar  {i}}-4{\bar  {j}}+2{\bar  {k}}\,,then write {\bar  {a}}\times {\bar  {b}}\,

solution Determine the unit vector perpendicular to both the vectors 2{\bar  {i}}+{\bar  {j}}+3{\bar  {k}},{\bar  {i}}-2{\bar  {j}}+{\bar  {k}}\,

solution Find the vector area of a parallelogram whose diagonals are determined by the vectors {\bar  {a}}=3{\bar  {i}}+{\bar  {j}}-2{\bar  {k}},{\bar  {b}}={\bar  {i}}-3{\bar  {j}}+4{\bar  {k}}\,

solution Find a vector of magnitude 3 and which is perpendicular to both the vectors 3{\bar  {i}}+{\bar  {j}}-4{\bar  {k}},6{\bar  {i}}+5{\bar  {j}}-2{\bar  {k}}\,

solution IF (1,2,3),(2,5,-1),(-1,1,2)\, are the vertices of a triangle, find its area.

solution Find a unit vector perpendiculars to the plane ABC where A=(3,-1,2),B(1,-1,-3),C(4,-3,1)\,

solution Let {\bar  {a}}=2{\bar  {i}}+3{\bar  {j}}+4{\bar  {k}},{\bar  {b}}={\bar  {i}}-2{\bar  {j}}+{\bar  {k}}\,.If a vector {\bar  {r}}\, satisfies {\bar  {a}}\times {\bar  {r}}=3{\bar  {b}}\, and {\bar  {a}}\cdot {\bar  {r}}=2\, then find the vector {\bar  {r}}\,

solution Find the volume of the parallelopiped whose edges are {\bar  {i}}+{\bar  {j}}+{\bar  {k}},{\bar  {i}}-{\bar  {j}}+{\bar  {k}},{\bar  {i}}+{\bar  {j}}-{\bar  {k}}\,.

solution Show that the four points having position vectors 6{\bar  {i}}-7{\bar  {j}},16{\bar  {i}}-19{\bar  {j}}-4{\bar  {k}},3{\bar  {i}}-6{\bar  {k}},2{\bar  {i}}+5{\bar  {j}}+10{\bar  {k}}\, are not coplanar.

solution If the two vectors {\bar  {a}}=2{\bar  {i}}+{\bar  {j}}+2{\bar  {k}},{\bar  {b}}=5{\bar  {i}}-3{\bar  {j}}+{\bar  {k}}\, are two vectors,find the projection of {\bar  {b}}\, on {\bar  {a}}\,

solution Reduce the equation {\bar  {r}}\cdot (4{\bar  {i}}-12{\bar  {j}}-3{\bar  {k}})+7=0\, to normal form and hence find the length of the perpendicular from the origin to the plane.

solution Find the angle between the planes {\bar  {r}}\cdot (2{\bar  {i}}-{\bar  {j}}+{\bar  {k}})=6,{\bar  {r}}\cdot ({\bar  {i}}+{\bar  {j}}+2{\bar  {k}})=7\,

solution Find the value of lambda for which the four points with position vectors 3{\bar  {i}}-2{\bar  {j}}-{\bar  {k}},2{\bar  {i}}+3{\bar  {j}}-4{\bar  {k}},-{\bar  {i}}+{\bar  {j}}+2{\bar  {k}},4{\bar  {i}}+5{\bar  {j}}+\lambda {\bar  {k}}\, are coplanar.

solution Find the volume of the tetrahedron with vertices (1,1,3),(4,3,2),(5,2,7),(6,4,8)\,

solution Find the vector equation of the line passing through three non-collinear points -2{\bar  {i}}+6{\bar  {j}}-6{\bar  {k}},-3{\bar  {i}}+10{\bar  {j}}-9{\bar  {k}},-5{\bar  {i}}-6{\bar  {k}}\,.Also find its cartesian equation.

solution Find the equation of the plane passing through the points 3{\bar  {i}}-5{\bar  {j}}-{\bar  {k}},-{\bar  {i}}+5{\bar  {j}}+7{\bar  {k}}\, and parallel to 3{\bar  {i}}-{\bar  {j}}+7{\bar  {k}}\,

solution If {\begin{vmatrix}a&a^{2}&1+a^{3}\\b&b^{2}&1+b^{3}\\c&c^{2}&1+c^{3}\end{vmatrix}}=0\, and the vectors {\bar  {A}}=(1,a,a^{2}),{\bar  {B}}=(1,b,b^{2}),{\bar  {C}}=(1,c,c^{2})\, are non-coplanar,then prove that abc=-1\,

solution Find the perpendicular distance from the origin to the plane passing through the points (1,-2,5),(0,-5,-1),(-3,5,0)\,

Jordan

solution {\begin{bmatrix}2&0&-1&1\\1&1&-1&1\\1&-3&2&-1\\0&-3&2&-1\end{bmatrix}}


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